Processing math: 52%
Research article

Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity

  • Received: 25 May 2021 Accepted: 16 August 2021 Published: 27 August 2021
  • MSC : 34F05, 60H10, 92D30

  • In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.

    Citation: Yuhuai Zhang, Xinsheng Ma, Anwarud Din. Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity[J]. AIMS Mathematics, 2021, 6(11): 12359-12378. doi: 10.3934/math.2021715

    Related Papers:

    [1] Yusuf Dogru . η-Ricci-Bourguignon solitons with a semi-symmetric metric and semi-symmetric non-metric connection. AIMS Mathematics, 2023, 8(5): 11943-11952. doi: 10.3934/math.2023603
    [2] Shahroud Azami, Mehdi Jafari, Nargis Jamal, Abdul Haseeb . Hyperbolic Ricci solitons on perfect fluid spacetimes. AIMS Mathematics, 2024, 9(7): 18929-18943. doi: 10.3934/math.2024921
    [3] Xiaosheng Li . New Einstein-Randers metrics on certain homogeneous manifolds arising from the generalized Wallach spaces. AIMS Mathematics, 2023, 8(10): 23062-23086. doi: 10.3934/math.20231174
    [4] Nasser Bin Turki, Sharief Deshmukh, Olga Belova . A note on closed vector fields. AIMS Mathematics, 2024, 9(1): 1509-1522. doi: 10.3934/math.2024074
    [5] Sharief Deshmukh, Mohammed Guediri . Some new characterizations of spheres and Euclidean spaces using conformal vector fields. AIMS Mathematics, 2024, 9(10): 28765-28777. doi: 10.3934/math.20241395
    [6] Amira Ishan . On concurrent vector fields on Riemannian manifolds. AIMS Mathematics, 2023, 8(10): 25097-25103. doi: 10.3934/math.20231281
    [7] Yanlin Li, Aydin Gezer, Erkan Karakaş . Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection. AIMS Mathematics, 2023, 8(8): 17335-17353. doi: 10.3934/math.2023886
    [8] Mohd Danish Siddiqi, Fatemah Mofarreh . Schur-type inequality for solitonic hypersurfaces in (k,μ)-contact metric manifolds. AIMS Mathematics, 2024, 9(12): 36069-36081. doi: 10.3934/math.20241711
    [9] Tong Wu, Yong Wang . Super warped products with a semi-symmetric non-metric connection. AIMS Mathematics, 2022, 7(6): 10534-10553. doi: 10.3934/math.2022587
    [10] Fahad Sikander, Tanveer Fatima, Sharief Deshmukh, Ayman Elsharkawy . Curvature analysis of concircular trajectories in doubly warped product manifolds. AIMS Mathematics, 2024, 9(8): 21940-21951. doi: 10.3934/math.20241066
  • In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.



    In recent years, a useful extension has been proposed from the classical calculus by permitting derivatives and integrals of arbitrary orders is known as fractional calculus. It emerged from a celebrated logical conversation between Leibniz and L'Hopital in 1695 and was enhanced by different scientists like Laplace, Abel, Euler, Riemann, and Liouville [1]. Fractional calculus has gained popularity on the account of diverse applications in various areas of science and technology [2,3,4]. The concept of this new calculus was applied in several distinguished areas previously with excellent developments in the frame of novel approaches and posted scholarly papers, see [5,6,7,8,9,10,11,12,13,14,15,16,17,18]. Various notable generalized fractional integral operators such as the Riemann-Liouville, Hadamard, Caputo, Marichev-Saigo-Maeda, Riez, the Gaussian hypergeometric operators and so on, their attempts helpful for researchers to recognize the real world phenomena. Therefore, the Caputo and Riemann-Liouville was the most used fractional operators having singular kernels. It is remarkable that all the above mentioned operators are the particular cases of the operators investigated by Jarad et al. [19]. The utilities to weighted generalized fractional operators are undertaking now.

    Adopting the excellency of the above work, we introduce a new weighted framework of generalized proportional fractional integral operator with respect to monotone function Ψ. Also, some new characteristics of the aforesaid operator are apprehended to explore new ideas to amplify the fractional operators and acquire fractional integral inequalities via generalized fractional operators (see Remark 2 and 3 below).

    Recently, by employing the fractional integral operators, several researchers have established a bulk of fractional integral inequalities and their variant forms with fertile applications. These sorts of speculations have noteworthy applications in fractional differential/difference equations and fractional Schrödinger equations [20,21]. By the use of Riemann-Liouville fractional integral operator, Belarbi and Dahmani [22] contemplated the subsequent integral inequalities as follows:

    If f1 and g1 are two synchronous functions on [0,), then

    Ωα(f1g1)(ϰ)Γ(α+1)ϰαΩα(f1)(ϰ)Ωα(g1)(ϰ) (1.1)

    and

    ϰαΓ(α+1)Ωβ(f1g1)(ϰ)+ϰβΓ(β+1)Ωα(f1g1)(ϰ)Ωα(f1)(ϰ)Ωβ(g1)(ϰ)+Ωβ(f1)(ϰ)Ωα(g1)(ϰ), (1.2)

    for all ϰ>0,α,β>0. Butt et al. [23], Rashid et al. [24] and Set et al. [25] established the fractional integral inequalities via generalized fractional integral operator having Raina's function, generalized K-fractional integral and Katugampola fractional integral inequalities similar to the variants (1.1) and (1.2), respectively. Here we should emphasize that, inequalities (1.1) and (1.2) are a remarkable instrument for reconnoitering plentiful scientific regions of investigation encompassing probability theory, statistical analysis, physics, meteorology, chaos and henceforth.

    More general version of inequalities (1.1) and (1.2) proposed by Dahmani [26] by employing Riemann-Liouville fractional integral operator.

    Let f1 and g1 be two synchronous functions on [0,) and let r,s:[0,)[0,). Then

    ΩαP(ϰ)Ωα(Qf1g1)(ϰ)+ΩαQ(ϰ)Ωα(Pf1g1)(ϰ)Ωα(Qf1)(ϰ)Ωα(Pg1)(ϰ)+Ωα(Pf1)(ϰ)Ωα(Qg1)(ϰ) (1.3)

    and

    ΩαP(ϰ)Ωβ(Qf1g1)(ϰ)+ΩβQ(ϰ)Ωα(Pf1g1)(ϰ)Ωα(Qf1)(ϰ)Ωβ(Pg1)(ϰ)+Ωβ(Pf1)(ϰ)Ωα(Qg1)(ϰ) (1.4)

    for all ϰ>0,α,β>0. Chinchane and Pachpatte [27], Brahim and Taf [28] and Shen et al. [29] explored the Hadamard fractional integral inequalities, the fractional version of integral inequalities in two variable quantum deformation and the Riemann-Liouville fractional integral operator on time scale analysis coincide to variants (1.3) and (1.4), respectively.

    Let us define the most distinguished Chebyshev functional [30]:

    T(f1,g1)=1b1a1b1a1f1(ϰ)g1(ϰ)dϰ1b1a1b1a1f1(ϰ)dϰ1b1a1b1a1g1(ϰ)dϰ, (1.5)

    where f1 and g1 are two integrable functions on [a1,b1]. In [31], Grüss proposed the well-known generalization:

    |T(f1,g1)|14(Φϕ)(Υγ), (1.6)

    where f1 and g1 are two integrable functions on [a1,b1] satisfying the assumptions

    ϕf1(ϰ)Φ,γg1(ϰ)Υ,ϕ,Φ,γ,ΥR,ϰ[a1,b1]. (1.7)

    The inequality (1.6) is known to be Grüss inequality. In recent years, the Grüss type integral inequality has been the subject of very active research. Mathematicians and scientists can see them in research papers, monographs, and textbooks devoted to the theory of inequalities [32,33,34,35] such as, Dragomir [36] demonstrated certain variants with the supposition of vectors and continuous mappings of selfadjoint operators in Hilbert space similar to (1.6). In this context, f1 and g1 are holding the assumptions (1.7), Dragomir [37] derived several functionals in two and three variable sense as follows:

    |S(f1,g1,P)|14(Φϕ)(Υγ)(b1a1P1(ϰ)dϰ)2, (1.8)

    where

    S(f1,g1,P)=12T(f1,g1,P)=b1a1P(ϰ)dϰb1a1P(ϰ)f1(ϰ)g1(ϰ)dϰb1a1P(ϰ)f1(ϰ)dϰb1a1P(ϰ)g1(ϰ)dϰ (1.9)

    and

    T(f1,g1,P,Q)=b1a1Q(ϰ)dϰb1a1P(ϰ)f1(ϰ)g1(ϰ)dϰ+b1a1P(ϰ)dϰb1a1Q(ϰ)f1(ϰ)g1(ϰ)dϰb1a1Q(ϰ)f1(ϰ)dϰb1a1P(ϰ)g1(ϰ)dϰb1a1P(ϰ)f1(ϰ)dϰb1a1Q(ϰ)g1(ϰ)dϰ. (1.10)

    In [37], Dragomir established the inequality:

    If f1,g1L(a1,b1), then

    |S(f1,g1,P)|f1g1(b1a1P(ϰ)dϰb1a1ϰ2P(ϰ)dϰ(b1a1ϰP(ϰ)dϰ)2). (1.11)

    Moreover, author [37] proved numerous variants for Lipschitzian functions as follows:

    If f1 is L-g1-Lipschitzian on [a1,b1], that is

    |f1(μ)fν|L|g1(μ)g1(ν)|,L>0,μ,ν[a1,b1]. (1.12)

    and

    |S(f1,g1,P)|L(b1a1P(ϰ)dϰb1a1g21(ϰ)P(ϰ)dϰ(b1a1g1(ϰ)P(ϰ)dϰ)2). (1.13)

    Furthermore, if f1 and g1 are L1 and L2-Lipschitzian functions on [a1,b1], then

    |S(f1,g1,P)|L1L2(b1a1P(ϰ)dϰb1a1ϰ2P(ϰ)dϰ(b1a1ϰP(ϰ)dϰ)2). (1.14)

    Owing to the above tendency, Dhamani et al. [38] proposed the fractional integral inequalities in the Riemann-Liouville parallel to variant (1.6) with the suppositions (1.7). Additionally, Dahamani and Benzidane [39] introduced weighted Grüss type inequality via (α,β)-fractional q-integral inequality resemble to (1.8) under the hypothesis of (1.5). Author [40,41] derived the extended functional of (1.10) by employing Riemann-Liouville integral corresponds to variants (1.11), (1.13) and (1.14), respectively. In this flow, Set et al. [42] contemplated the Grüss type inequalities considering the generalized K-fractional integral. Chen et al. [43] obtained the novel refinements of Hermite-Hadamard type inequalities for n-polynomial p-convex functions within the generalized fractional integral operators. Abdeljawad et al. [44] derived the Simpson's type inequalities for generalized p-convex functions involving fractal set. Jarad et al. [45] investigated the properties of the more general form of generalized proportional fractional operators in Laplace transforms.

    The motivation of this paper is twofold. First, we propose a novel framework named weighted generalized proportional fractional integral operator based on characteristics, as well as considering the boundedness and semi-group property and able to be widely applied to many scientific results. Second, the current operator employed to the extended weighted Chebyshev and Grüss type inequalities for exploring the analogous versions of (1.5) and (1.6). Some special cases are pictured with new fractional operators which are not computed yet. Interestingly, particular cases are designed for Riemann-Liouville fractional integral, generalized Riemann-Liouville fractional integral and generalized proportional fractional integral inequalities. It is worth mentioning that these operators have the ability to recapture several generalizations in the literature by considering suitable assumptions of Ψ,ω and ρ.

    In this section, we demonstrate the space where the weighted fractional integrals are bounded and also, provide certain specific features of these operators.

    Definition 2.1 ([19])Let ω0 be a mapping defined on [a1,b1], g1 is a differentiable strictly increasing function on [a1,b1]. The space χpω(a1,b1),1p< is the space of all Lebesgue measurable functions f1 defined on [a1,b1] for which f1χpω, where

    f1χpω=(b1a1|ω(ϰ)f1(ϰ)|pg1(ϰ)dϰ)1p,1<p< (2.1)

    and

    f1χpω=esssupa1ϰb1|ω(ϰ)f1(ϰ)|<. (2.2)

    Remark 1. Clearly we see that f1χpω(a1,b1) ω(ϰ)f1(ϰ)(g11(ϰ))1/pLp(a1,b1) for 1p< and f1χpω(a1,b1) ω(ϰ)f1(ϰ)L(a1,b1).

    Now, we show a novel fractional integral operator which is known as the weighted generalized proportional fractional integral operator with respect to monotone function Ψ.

    Definition 2.2. Let f1χpω(a1,b1) and ω0 be a function on [a1,b1]. Also, assume that Ψ is a continuously differentiable function on [a1,b1] with ψ>0 on [a1,b1]. Then the left and right-sided weighted generalized proportional fractional integral operator with respect to another function Ψ of order α>0 are described as:

    ΨωΩρ;αa1f1(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αf1(μ)ω(μ)Ψ(μ)dμ,a1<ϰ (2.3)

    and

    ΨωΩρ;αb1f1(ϰ)=ω1(ϰ)ραΓ(α)b1ϰexp[ρ1ρ(Ψ(μ)Ψ(ϰ))](Ψ(μ)Ψ(ϰ))1αf1(μ)ω(μ)Ψ(μ)dμ,ϰ<b1, (2.4)

    where ρ(0,1] is the proportionality index, αC,(α)>0 and Γ(ϰ)=0μϰ1eμdμ is the Gamma function.

    Remark 2. Some particular fractional operators are the special cases of (2.3) and (2.4).

    (1) Setting Ψ(ϰ)=ϰ, in Definition (2.2), then we get the weighted generalized proportional fractional operators stated as follows:

    ωΩρ;αa1f1(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(ϰμ)](ϰμ)1αf1(μ)ω(μ)dμ,a1<ϰ (2.5)

    and

    ωΩρ;αb1f1(ϰ)=ω1(ϰ)ραΓ(α)b1ϰexp[ρ1ρ(μϰ)](μϰ)1αf1(μ)ω(μ)dμ,ϰ<b1. (2.6)

    (2) Setting Ψ(ϰ)=ϰ and ρ=1 in Definition (2.2), then we get the weighted Riemann-Liouville fractional operators stated as follows:

    ωΩαa1f1(ϰ)=ω1(ϰ)Γ(α)ϰa1f1(μ)ω(μ)dμ(ϰμ)1α,a1<ϰ (2.7)

    and

    ωΩαb1f1(ϰ)=ω1(ϰ)Γ(α)b1ϰf1(μ)ω(μ)dμ(μϰ)1α,ϰ<b1. (2.8)

    (3) Setting Ψ(ϰ)=lnϰ and a1>0 in Definition (2.2), we get the weighted generalized proportional Hadamard fractional operators stated as follows:

    ωΩρ;αa1f1(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(lnϰμ)](lnϰμ)1αf1(μ)ω(μ)μdμ,a1<ϰ (2.9)

    and

    ωΩρ;αb1f1(ϰ)=ω1(ϰ)ραΓ(α)b1ϰexp[ρ1ρ(lnμϰ)](lnμϰ)1αf1(μ)ω(μ)μdμ,ϰ<b1. (2.10)

    (4) Setting Ψ(ϰ)=lnϰ and a1>0 along with ρ=1 in Definition (2.2), then we get the weighted Hadamard fractional operators stated as follows:

    ωΩαa1f1(ϰ)=ω1(ϰ)Γ(α)ϰa1f1(μ)ω(μ)dμμ(lnϰμ)1α,a1<ϰ (2.11)

    and

    ωΩαb1f1(ϰ)=ω1(ϰ)Γ(α)b1ϰf1(μ)ω(μ)dμμ(lnμϰ)1α,ϰ<b1. (2.12)

    (5) Setting Ψ(ϰ)=ϰττ(τ>0) in Definition (2.2), then we get the weighted generalized fractional operators in terms of Katugampola stated as follows:

    ωΩαa1f1(ϰ)=ω1(ϰ)Γ(α)ϰa1(ϰτμττ)α1f1(μ)ω(μ)dμμ1τ,a1<ϰ (2.13)

    and

    ωΩαb1f1(ϰ)=ω1(ϰ)Γ(α)b1ϰ(μτϰττ)α1f1(μ)ω(μ)dμμ1τ,ϰ<b1. (2.14)

    Remark 3. Several existing integral operators can be derived from Definition 2.2 as follows:

    (1) Letting ω(ϰ)=1, then we get the Definition 4 proposed by Rashid et al. [46] and Definition 3.2 introduced by Jarad et al. [47], independently.

    (2) Letting ω(ϰ)=1,Ψ(ϰ)=ϰ, then we get the Definition 3.4 defined by Jarad et al. [48].

    (3) Letting ω(ϰ)=1 and Ψ(ϰ)=lnϰ along with a1>0, then we get the Definition 2.1 defined by Rahman et al. [49].

    (4) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=lnϰ along with a1>0, then we get the operator defined by Kilbas et al. [3] and Smako et al. [5], respectively.

    (5) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=ϰ, then we get the operator defined by Kilbas et al [3].

    (6) Letting ω(ϰ)=1 and Ψ(ϰ)=ϰττ,(τ>0), then we get the operator defined by Katugampola et al. [7].

    (7) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=ϰτ+sτ+s,τ(0,1],sR, then we get the Definition 2 defined by Khan and Khan et al [50].

    (8) Letting ω(ϰ)=ρ=1 and Ψ(ϰ)=(ϰa1)ττ, and Ψ(ϰ)=(b1ϰ)ττ,(τ>0), then we get the operator defined by Jarad et al. [51].

    Theorem 2.3. For α>0,ρ(0,1],1p and f1χpω(a1,b1). Then ΨωΩρ;αa1 is bounded in χpω(a1,b1) and

    ΨωΩρ;αa1f1χpω(Ψ(b1)Ψ(a1))αf1χpωραΓ(α+1).

    Proof. For 1p, we have

    ΨωΩρ;αa1f1χpω=1ραΓ(α)(b1a1|ϰa1exp[ρ1ρΨ(ϰ)Ψ(μ)](Ψ(ϰ)Ψ(μ))1αω(μ)f1(μ)Ψ(μ)dμ|pΨ(ϰ)dϰ)1/p=1ραΓ(α)(Ψ(b1)Ψ(a1)|t2Ψ(a1)exp[ρ1ρ(t2t1)](t2t1)1αω(Ψ1(t1))f1(Ψ1(t1))|pdt2)1/p.

    Using the fact that |exp[ρ1ρ(t2t1)]|<1. Taking into account the generalized Minkowski inequality [5], we can write

    ΨωΩρ;αa1f1χpω1ραΓ(α)Ψ(b1)Ψ(a1)(|ω(Ψ1(t1))f1(Ψ1(t1))|pΨ(b1)t1(t2t1)p(α1)dt2)1/pdt1=1ραΓ(α)Ψ(b1)Ψ(a1)(|ω(Ψ1(t1))f1(Ψ1(t1))|((Ψ(b1)t1)p(α1)+1p(α1)+1)1/pdt1.

    By employing the well-known Hölder inequality satisfying p1+q1=1, we obtain

    ΨωΩρ;αa1f1χpω1ραΓ(α)(Ψ(b1)Ψ(a1)|ω(Ψ1(t1))f1(Ψ1(t1))|pdt1)1/p(Ψ(b1)Ψ(a1)((Ψ(b1)t1)p(α1)+1p(α1)+1)q/pdt1)1/q1ραΓ(α)(b1a1|ω(ϰ)f1(ϰ)|pΨ(ϰ)dϰ)1/p(Ψ(b1)Ψ(a1)((Ψ(b1)t1)p(α1)+1p(α1)+1)q/pdt1)1/q(Ψ(b1)Ψ(a1))αf1χpωραΓ(α+1).

    Now, for p=, we have

    |ω(ϰ)ΨωΩρ;αa1f1(ϰ)|=1ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αf1(μ)ω(μ)Ψ(μ)dμ1ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1α|f1(μ)ω(μ)|Ψ(μ)dμ,Since(|exp[ρ1ρ(t2t1)]|<1)f1χωραΓ(α)ϰa1(Ψ(ϰ)Ψ(μ))α1dμ(Ψ(ϰ)Ψ(a1))αf1χωραΓ(α+1)=(Ψ(b1)Ψ(a1))αf1χωραΓ(α+1).

    This ends the proof.

    Our next result is the semi group property for weighted generalized proportional fractional integral operator with respect to monotone function.

    Theorem 2.4. For α,β>0,ρ(0,1] with 1p and let f1χpω(a1,b1). Then

    (ΨωΩρ;αa1ΨωΩρ;βa1)f1=(ΨωΩρ;α+βa1)f1. (2.15)

    Proof.

    (ΨωΩρ;αa1ΨωΩρ;βa1f1)(ϰ)=ω1(ϰ)ραΓ(α)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αω(μ)(ΨωΩρ;βa1f1)(μ)Ψ(μ)dμ=ω1(ϰ)ρα+βΓ(α)Γ(β)ϰa1μa1exp[ρ1ρ(Ψ(ϰ)Ψ(μ))](Ψ(ϰ)Ψ(μ))1αexp[ρ1ρ(Ψ(μ)Ψ(ν))](Ψ(μ)Ψ(ν))1β×ω(ν)f1(ν)Ψ(ν)Ψ(μ)dμdν.

    By making change of variable technique θ=Ψ(μ)Ψ(a1)Ψ(ϰ)Ψ(a1), we can write

    (ΨωΩρ;αa1ΨωΩρ;βa1f1)(ϰ)=ω1(ϰ)ρα+βΓ(α)Γ(β)10θβ1(1θ)α1dθϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(ν))](Ψ(ϰ)Ψ(ν))1αβω(ν)f1(ν)Ψ(ν)dν=ω1(ϰ)ρα+βΓ(α)Γ(β)Γ(α)Γ(β)Γ(α+β)ϰa1exp[ρ1ρ(Ψ(ϰ)Ψ(ν))](Ψ(ϰ)Ψ(ν))1αβω(ν)f1(ν)Ψ(ν)dν=(ΨωΩρ;α+βa1f1)(ϰ),

    where B(α,β)=Γ(α)Γ(β)Γ(α+β)=10θβ1(1θ)α1dθ is known to be Euler Beta function.

    This section contains some significant generalizations for weighted integral inequalities by employing weighted generalized proportional fractional integral operator, for the consequences relating to (1.1) and (1.2), it is suppose that all mappings are integrable in the Riemann sense.

    Throughout this investigation, we use the following assumptions:

    I. Let f1 and g1 be two synchronous functions on [0,).

    II. Let Ψ:[0,)(0,) is an increasing function with continuous derivative Ψ on the interval (0,).

    Lemma 3.1. If the supposition I and II are satisfied and let Q and P be two non-negative continuous mappings on [0,). Then the inequality

    ΨωΩρ;α0+(P)(ϰ)ΨωΩρ;α0+(Qf1g1)(ϰ)+ΨωΩρ;α0+(Pf1g1)(ϰ)ΨωΩρ;α0+(Q)(ϰ)ΨωΩρ;α0+(Pg1)(ϰ)ΨωΩρ;α0+(Qf1)(ϰ)+ΨωΩρ;α0+(Pf1)(ϰ)ΨωΩρ;α0+(Qg1)(ϰ), (3.1)

    holds for all ρ(0,1],αC with (α)>0.

    Proof. Since f1 and g1 are two synchronous functions on [0,), then for all μ>0 and ν>0, we have

    (f1(μ)f1(ν))(g1(μ)g1(ν))0. (3.2)

    By (3.2), we write

    f1(μ)g1(μ)+f1(ν)g1(ν)g1(μ)f1(ν)+g1(ν)f1(μ). (3.3)

    If we multiply both sides of (3.3) by exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]Q(μ)ω(μ)Ψ(μ)ραΓ(α)(Ψ(ϰ)Ψ(μ))1α and integrating the resulting inequality with respect to μ from 0 to ϰ, we get

    1ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]Q(μ)ω(μ)Ψ(μ)ραΓ(α)(Ψ(ϰ)Ψ(μ))1αf1(μ)g1(μ)dμ+f1(ν)g1(ν)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]Q(μ)ω(μ)Ψ(μ)ραΓ(α)(Ψ(ϰ)Ψ(μ))1αdμf1(ν)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]Q(μ)ω(μ)Ψ(μ)ραΓ(α)(Ψ(ϰ)Ψ(μ))1αg1(ν)dν+g1(ν)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]Q(μ)ω(μ)Ψ(μ)ραΓ(α)(Ψ(ϰ)Ψ(μ))1αf1(μ)dμ. (3.4)

    Taking product both sides of the above equation by ω1(ϰ) and in view of Definition (2.2), we have

    ΨωΩρ;α0+(Qf1g1)(ϰ)+f1(ν)g1(ν)ΨωΩρ;α0+(Q)(ϰ)g1(ν)ΨωΩρ;α0+(Qf1)(ϰ)+f1(ν)ΨωΩρ;α0+(Qg1)(ϰ). (3.5)

    Further, if we multiply both sides of (3.5) by exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]P(ν)ω(ν)Ψ(ν)ραΓ(α)(Ψ(ϰ)Ψ(ν))1α and integrating the resulting inequality with respect to ν from 0 to ϰ. Then, multiplying by ω1(ϰ) and in view of Definition 2.2, we obtain

    ΨωΩρ;α0+(P)(ϰ)ΨωΩρ;α0+(Qf1g1)(ϰ)+ΨωΩρ;α0+(Pf1g1)(ϰ)ΨωΩρ;α0+(Q)(ϰ)ΨωΩρ;α0+(Pg1)(ϰ)ΨωΩρ;α0+(Qf1)(ϰ)+ΨωΩρ;α0+(Pf1)(ϰ)ΨωΩρ;α0+(Qg1)(ϰ), (3.6)

    which implies (3.1).

    Theorem 3.2. Under the assumption of I, II and let r, s and t be three non-negative continuous functions on [0,). Then the inequality

    2ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+s(ϰ)ΨωΩρ;α0+(tf1g1)(ϰ)+ΨωΩρ;α0+(sf1g1)(ϰ)ΨωΩρ;α0+t(ϰ))+2ΨωΩρ;α0+(rf1g1)(ϰ)ΨωΩρ;α0+s(ϰ)ΨωΩρ;α0+t(ϰ)ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;α0+(tf1)(ϰ)+ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;α0+(tg1)(ϰ))+ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;α0+(tf1)(ϰ)+ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;α0+(tg1)(ϰ))+ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;α0+(rf1)(ϰ)+ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;α0+(rg1)(ϰ)) (3.7)

    holds for all ρ(0,1],αC with (α)>0.

    Proof. By means of Lemma 3.1 and setting P=r,Q=s, we can write

    ΨωΩρ;α0+s(ϰ)ΨωΩρ;α0+(tf1g1)(ϰ)+ΨωΩρ;α0+(sf1g1)(ϰ)ΨωΩρ;α0+t(ϰ)ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;α0+(tf1)(ϰ)+ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;α0+(tg1)(ϰ). (3.8)

    Conducting product both sides of (3.8) by ΨωΩρ;α0+r(ϰ), we obtain

    ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+s(ϰ)ΨωΩρ;α0+(tf1g1)(ϰ)+ΨωΩρ;α0+(sf1g1)(ϰ)ΨωΩρ;α0+t(ϰ))ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;α0+(tf1)(ϰ)+ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;α0+(tg1)(ϰ)). (3.9)

    By means of Lemma 3.1 and setting P=r,Q=t, we can write

    ΨωΩρ;α0+r(ϰ)ΨωΩρ;α0+(tf1g1)(ϰ)+ΨωΩρ;α0+(rf1g1)(ϰ)ΨωΩρ;α0+t(ϰ)ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;α0+(tf1)(ϰ)+ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;α0+(tg1)(ϰ). (3.10)

    Conducting product of (3.10) by ΨωΩρ;α0+s(ϰ), we obtain

    ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+r(ϰ)ΨωΩρ;α0+(tf1g1)(ϰ)+ΨωΩρ;α0+(rf1g1)(ϰ)ΨωΩρ;α0+t(ϰ))ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;α0+(tf1)(ϰ)+ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;α0+(tg1)(ϰ)). (3.11)

    By similar argument as we did before, yields

    ΨωΩρ;α0+t(ϰ)(ΨωΩρ;α0+r(ϰ)ΨωΩρ;α0+(sf1g1)(ϰ)+ΨωΩρ;α0+(rf1g1)(ϰ)ΨωΩρ;α0+t(ϰ))ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;α0+(rf1)(ϰ)+ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;α0+(rg1)(ϰ)). (3.12)

    Adding (3.9), (3.11) and (3.12), we get the desired inequality (3.8).

    Lemma 3.3. Under the assumption of I, II and let Q and P be two non-negative continuous functions on [0,). Then the inequality

    ΨωΩρ;α0+(Pf1g1)(ϰ)ΨωΩρ;β0+Q(ϰ)+ΨωΩρ;α0+P(ϰ)ΨωΩρ;β0+(Qf1g1)(ϰ)ΨωΩρ;α0+(Pf1)(ϰ)ΨωΩρ;β0+(Qg1)(ϰ)+ΨωΩρ;α0+(Pg1)(ϰ)ΨωΩρ;β0+(Qf1)(ϰ),

    holds for all ρ(0,1],α,βC with (α),(β)>0.

    Proof. If we multiply both sides of (3.2) by exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]Q(ν)ω(ν)Ψ(ν)ρβΓ(β)(Ψ(ϰ)Ψ(ν))1β and integrating the resulting inequality with respect to ν from 0 to ϰ, we have

    f1(μ)g1(μ)ρβΓ(β)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]Q(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1βdν+f1(ν)g1(ν)ρβΓ(β)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]Q(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1βdνg1(μ)ρβΓ(β)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]Q(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1βf1(ν)dν+f1(μ)ρβΓ(β)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]Q(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1βg1(ν)dν. (3.13)

    Taking product both sides of the above equation by ω1(ϰ) and in view of Definition (2.2), we have

    f1(μ)g1(μ)ΨωΩρ;β0+Q(ϰ)+ΨωΩρ;β0+(Qf1g1)(ϰ)f1(μ)ΨωΩρ;β0+(Qg1)(ϰ)+g1(μ)ΨωΩρ;β0+(Qf1)(ϰ). (3.14)

    Again, multiplying both sides of (3.14) by exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]P(μ)ω(μ)Ψ(μ)ραΓ(α)(Ψ(ϰ)Ψ(μ))1α and integrating the resulting inequality with respect to ν from 0 to ϰ, we have

    ΨωΩρ;β0+Q(ϰ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]P(μ)ω(μ)Ψ(μ)(Ψ(ϰ)Ψ(μ))1αf1(μ)g1(μ)dμ+ΨωΩρ;β0+(Qf1g1)(ϰ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]P(μ)ω(μ)Ψ(μ)(Ψ(ϰ)Ψ(μ))1αdμΨωΩρ;β0+(Qg1)(ϰ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]P(μ)ω(μ)Ψ(μ)(Ψ(ϰ)Ψ(μ))1αf1(μ)dμ+ΨωΩρ;β0+(Qf1)(ϰ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(μ))]P(μ)ω(μ)Ψ(μ)(Ψ(ϰ)Ψ(μ))1αg1(μ)dμ. (3.15)

    Taking product both sides of the above equation by ω1(ϰ) and in view of Definition (2.2), we obtain

    ΨωΩρ;α0+(Pf1g1)(ϰ)ΨωΩρ;β0+Q(ϰ)+ΨωΩρ;α0+P(ϰ)ΨωΩρ;β0+(Qf1g1)(ϰ)ΨωΩρ;α0+(Pf1)(ϰ)ΨωΩρ;β0+(Qg1)(ϰ)+ΨωΩρ;α0+(Pg1)(ϰ)ΨωΩρ;β0+(Qf1)(ϰ),

    which implies (3.13).

    Theorem 3.4. Under the assumptions I, II and let r, s and t be three non-negative continuous functions on [0,). Then the inequality

    ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+(sf1g1)(ϰ)ΨωΩρ;β0+t(ϰ)+2ΨωΩρ;α0+s(ϰ)ΨωΩρ;β0+(tf1g1)(ϰ)+ΨωΩρ;β0+t(ϰ)ΨωΩρ;α0+(sf1g1)(ϰ))+(ΨωΩρ;β0+t(ϰ)ΨωΩρ;α0+s(ϰ)+ΨωΩρ;α0+t(ϰ)ΨωΩρ;β0+s(ϰ))ΨωΩρ;α0+(rf1g1)(ϰ)ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;β0+(tg1)(ϰ)+ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;β0+(tf1)(ϰ))+ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;β0+(tg1)(ϰ)+ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;β0+(tf1)(ϰ))+ΨωΩρ;α0+t(ϰ)(ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;β0+(sg1)(ϰ)+ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;β0+(sf1)(ϰ)) (3.16)

    holds for all ρ(0,1],α,βC with (α),(β)>0.

    Proof. By means of Lemma 3.3 and setting P=s,Q=t, we can write

    ΨωΩρ;α0+(sf1g1)(ϰ)ΨωΩρ;β0+t(ϰ)+ΨωΩρ;α0+s(ϰ)ΨωΩρ;β0+(tf1g1)(ϰ)ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;β0+(tg1)(ϰ)+ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;β0+(tf1)(ϰ). (3.17)

    Conducting product both sides of (3.17) by ΨωΩρ;α0+r(ϰ), we obtain

    ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+(sf1g1)(ϰ)ΨωΩρ;β0+t(ϰ)+ΨωΩρ;α0+s(ϰ)ΨωΩρ;β0+(tf1g1)(ϰ))ΨωΩρ;α0+r(ϰ)(ΨωΩρ;α0+(sf1)(ϰ)ΨωΩρ;β0+(tg1)(ϰ)+ΨωΩρ;α0+(sg1)(ϰ)ΨωΩρ;β0+(tf1)(ϰ)). (3.18)

    Again, by means of Lemma 3.3 and setting P=r,Q=t, we can write

    ΨωΩρ;α0+(rf1g1)(ϰ)ΨωΩρ;β0+t(ϰ)+ΨωΩρ;α0+r(ϰ)ΨωΩρ;β0+(tf1g1)(ϰ)ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;β0+(tg1)(ϰ)+ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;β0+(tf1)(ϰ). (3.19)

    Conducting product both sides of (3.19) by ΨωΩρ;α0+s(ϰ), we obtain

    ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+(rf1g1)(ϰ)ΨωΩρ;β0+t(ϰ)+ΨωΩρ;α0+r(ϰ)ΨωΩρ;β0+(tf1g1)(ϰ))ΨωΩρ;α0+s(ϰ)(ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;β0+(tg1)(ϰ)+ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;β0+(tf1)(ϰ)). (3.20)

    By similar arguments as we did before, yields

    ΨωΩρ;α0+t(ϰ)(ΨωΩρ;α0+(sf1g1)(ϰ)ΨωΩρ;β0+r(ϰ)+ΨωΩρ;α0+s(ϰ)ΨωΩρ;β0+(rf1g1)(ϰ))ΨωΩρ;α0+t(ϰ)(ΨωΩρ;α0+(rf1)(ϰ)ΨωΩρ;β0+(sg1)(ϰ)+ΨωΩρ;α0+(rg1)(ϰ)ΨωΩρ;β0+(sf1)(ϰ)). (3.21)

    Adding (3.18), (3.20) and (3.21), we get the desired inequality (3.16).

    Remark 4. Theorem 3.2 and Theorem 3.4 lead to the following conclusions:

    (1) Let f1 and g1 are the asynchronous functions on [0,), then (3.8) and (3.16) are reversed.

    (2) Let r,s and t are negative on [0,), then (3.8) and (3.16) are reversed.

    (3) Let r,s are positive t is negative on [0,), then (3.8) and (3.16) are reversed.

    In the next, we derive certain novel Grüss-type integral inequalities via weighted generalized proportional fractional integral operators.

    Lemma 3.5. Suppose an integrable function f1 defined on [0,) satisfying the assertions I,II and (1.7) on [0,) and let a continuous function r defined on [0,). Then the inequality

    ΨωΩρ;α0+r(ϰ)ΨωΩρ;α0+(rf21)(ϰ)(ΨωΩρ;α0+(rf1)(ϰ))2(ΦΨωΩρ;α0+x(ϰ)ΨωΩρ;α0+(rf1)(ϰ))(ΨωΩρ;α0+(rf1)(ϰ)ϕΨωΩρ;α0+r(ϰ))ΨωΩρ;α0+r(ϰ)ΨωΩρ;α0+(r(ϰ)(Φf1(ϰ))(f1(ϰ)ϕ)) (3.22)

    holds for all ρ(0,1],αC with (α)>0.

    Proof. By the given hypothesis and utilizing (1.7). For any μ,ν[0,), we have

    (Φf1(ν))(f1(μ)ϕ)+(Φf1(μ))(f1(ν)ϕ)(Φf1(μ))(f1(μ)ϕ)(Φf1(ν))(f1(ν)ϕ)f21(μ)+f21(ν)2f1(μ)f1(ν). (3.23)

    Multiplying both sides of (3.23) by exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)ραΓ(α)(Ψ(ϰ)Ψ(ν))1α and integrating the resulting inequality with respect to ν from 0 to ϰ, we have

    (f1(μ)ϕ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1α(Φf1(ν))dν+(Φf1(μ))ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1α(f1(ν)ϕ)dν(Φf1(μ))(f1(μ)ϕ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1αdν1ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1α(Φf1(ν))(f1(ν)ϕ)dνf21(μ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1αdν+1ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1αf21(ν)dν2f1(μ)ραΓ(α)ϰ0exp[ρ1ρ(Ψ(ϰ)Ψ(ν))]r(ν)ω(ν)Ψ(ν)(Ψ(ϰ)Ψ(ν))1αf1(ν)dν. (3.24)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we obtain

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(f_{1}(\mu)-\phi\big)+\big(\Phi-f_{1}(\mu)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r(\varkappa)(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\big)\\&&\leq f_{1}^{2}(\mu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)-2f_{1}(\mu)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa). \end{eqnarray} (3.25)

    Multiplying both sides of (3.25) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}} and integrating the resulting inequality with respect to \mu from 0 to \varkappa, we have

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\nu)\big)\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\big(f_{1}(\mu)-\phi\big)d\mu\\&&\quad+\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\big(\Phi-f_{1}(\mu)\big)d\mu\\&&\quad-\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r(\varkappa)(\Phi-f_{1}(\nu)\big)\big(f_{1}(\nu)-\phi\big)\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}d\nu\\&&\leq\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}^{2}(\mu)d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad+\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\\&&\quad-2\bigg(\frac{1}{\rho^{\alpha}\Gamma(\alpha)}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}f_{1}(\mu)d\mu\bigg)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa). \end{eqnarray} (3.26)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we obtain

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad+\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(r(\varkappa)(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)\big(\Phi-f_{1}(\varkappa)\big)(f_{1}(\varkappa)-\phi)\Big)\\&&\leq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa), \end{eqnarray} (3.27)

    which gives (3.22) and proves the lemma.

    Theorem 3.6. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let a continuous function r defined on [0, \infty) . Then the inequality

    \begin{eqnarray} \Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\leq\frac{(\Phi-\phi)(\Upsilon-\gamma)}{4}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)^{2} \end{eqnarray} (3.28)

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. By the given hypothesis stated in Theorem 3.6. Also, assume that \mathfrak{\mu, \nu} be defined by

    \begin{eqnarray} \mathfrak{T}(\mu, \nu) = \big(f_{1}(\mu)-f_{1}(\nu)\big)\big(g_{1}(\mu)-g_{1}(\nu)\big), \quad\mu, \nu\in[0, \varkappa], \quad\varkappa > 0. \end{eqnarray} (3.29)

    Multiplying both sides of (3.30) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} and integrating the resulting inequality with respect to \mu and \nu from 0 to \varkappa, we can state that

    \begin{eqnarray} &&\frac{1}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = \frac{1}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}}\\&&\quad\times\big(f_{1}(\mu)-f_{1}(\nu)\big)\big(g_{1}(\mu)-g_{1}(\nu)\big)d\mu d\nu. \end{eqnarray} (3.30)

    Taking product both sides of the above equation by \omega^{-1}(\varkappa) and in view of Definition (2.2), we obtain

    \begin{eqnarray} &&\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = 2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa). \end{eqnarray} (3.31)

    Thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, we can write that

    \begin{eqnarray} &&\Bigg(\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \mathfrak{T}(\mu, \nu)d\mu d\nu\Bigg)^{2}\\&&\leq \bigg(\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)} \int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \big(f_{1}(\mu)-f_{1}(\nu)\big)d\mu d\nu\bigg)\\&&\quad\bigg(\frac{\omega^{-2}(\varkappa)}{\rho^{2\alpha}\Gamma^{2}(\alpha)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]r(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\alpha}} \big(g_{1}(\mu)-g_{1}(\nu)\big)d\mu d\nu\bigg)\\&& = 4\bigg(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)^{2}\bigg)\\&&\quad\times\bigg(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\bigg). \end{eqnarray} (3.32)

    Since \big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)\geq0 and \big(\Upsilon-g_{1}(\mu)\big)\big(g_{1}(\mu)-\gamma\big)\geq0, we have

    \begin{eqnarray} \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)\big(\Phi-f_{1}(\mu)\big)\big(f_{1}(\mu)-\phi\big)\Big)\geq0, \end{eqnarray} (3.33)

    and

    \begin{eqnarray} \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)\big(\Upsilon-g_{1}(\mu)\big)\big(g_{1}(\mu)-\gamma\big)\Big)\geq0. \end{eqnarray} (3.34)

    Therefore, from (3.33), (3.34) and Lemma 3.5, we get

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big) \end{eqnarray} (3.35)

    and

    \begin{eqnarray} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)-\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big). \end{eqnarray} (3.36)

    Combining (3.30), (3.31), (3.35) and (3.36), we deduce that

    \begin{eqnarray} &&\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(xf_{1}g_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf)(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)\\&&\quad\times\Big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big). \end{eqnarray} (3.37)

    Taking into consideration the elementary inequality 4a_{1}a_{2}\leq(a_{1}+a_{2})^{2}, \, a_{1}, a_{2}\in\mathbb{R}, we can state that

    \begin{eqnarray} 4\Big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)\leq\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)(\Phi-\phi)\Big)^{2} \end{eqnarray} (3.38)

    and

    \begin{eqnarray} 4\Big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big)\leq\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)(\Upsilon-\gamma)\Big)^{2}. \end{eqnarray} (3.39)

    From (3.37)-(3.39), we obtain (3.28). This completes the proof of Theorem 3.6.

    Lemma 3.7. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} && \Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\Big)\\&&\quad\times\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\Big)\\ \end{eqnarray} (3.40)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Taking product (3.30) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\alpha}\Gamma(\alpha)(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{\rho^{\beta}\Gamma(\beta)(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}} and integrating the resulting inequality with respect to \mu and \nu from 0 to \varkappa, we can state that

    \begin{eqnarray} &&\frac{1}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = \frac{1}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\\&&\quad\times\big(f_{1}(\mu)-f_{1}(\nu)\big)\big(g_{1}(\mu)-g_{1}(\nu)\big)d\mu d\nu. \end{eqnarray} (3.41)

    Taking product both sides of the above equation by \omega^{-2}(\varkappa) and utilizing Definition (2.2), we have

    \begin{eqnarray} &&\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&& = \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa). \end{eqnarray} (3.42)

    Then, thanks to the weighted Cauchy-Schwartz integral inequality for double integrals, we conclude (3.40).

    Lemma 3.8. Suppose an integrable function f_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I} and \boldsymbol{II} on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} && \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\\&&\leq\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)\big(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\Big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Phi-f_{1}(\varkappa))\big(f_{1}(\varkappa)-\phi\big)\Big) \end{eqnarray} (3.43)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Multiplying both sides of (3.25) by \frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{\rho^{\beta}\Gamma(\beta)(\Psi(\varkappa)-\Psi(\mu))^{1-\beta}} and integrating the resulting inequality with respect to \mu from 0 to \varkappa. Then, by multiplying with \omega^{-1}(\varkappa) and in view of Definition 2.2, concludes

    \begin{eqnarray} &&\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)\big(\Phi-f_{1}(\varkappa)\big)\big(f_{1}(\varkappa)-\phi\big)\Big)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Phi-f_{1}(\varkappa))\big(f_{1}(\varkappa)-\phi\big)\Big)\\&&\leq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}^{2})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa), \end{eqnarray} (3.44)

    which gives (3.43) and proves the lemma.

    Theorem 3.9. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} && \Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big)^{2}\\&&\leq\Big\{\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)-\phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\big(\Phi\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1})(\varkappa)\big)\Big\}\\&&\quad\times\Big\{\big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\big)\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\big)\\&&\quad+\big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)-\gamma\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\big)\big(\Upsilon\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\big)\Big\} \end{eqnarray} (3.45)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Since (\Phi-f_{1}(\mu))(f_{1}(\mu)-\phi)\geq0 and (\Upsilon-g_{1}(\mu))(g_{1}(\mu)-\gamma)\geq0, we have

    \begin{eqnarray} -\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)(\Phi-f_{1}(\varkappa))(f_{1}(\varkappa)-\phi)\Big)- \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Phi-f_{1}(\varkappa))(f_{1}(\varkappa)-\phi)\Big)\leq0 \end{eqnarray} (3.46)

    and

    \begin{eqnarray} -\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}\Big(s(\varkappa)(\Upsilon-g_{1}(\varkappa))(g_{1}(\varkappa)-\gamma)\Big)- \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\Big(r(\varkappa)(\Upsilon-g_{1}(\varkappa))(g_{1}(\varkappa)-\gamma)\Big)\leq0. \end{eqnarray} (3.47)

    Utilizing Lemma 3.8 to f_{1} and g_{1}, and utilizing Lemma 3.7 and the inequalities (3.46) and (3.47), yields (3.45).

    Theorem 3.10. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)(\Phi-\phi)(\Upsilon-\gamma) \end{eqnarray} (3.48)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Taking into consideration the assumption (1.7), we have

    \begin{eqnarray} \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\leq \Phi-\phi, \quad\quad \Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq \Upsilon-\gamma, \quad\mu, \nu\in[0, \infty), \end{eqnarray} (3.49)

    which implies that

    \begin{eqnarray} \big\vert\mathfrak{T}(\mu, \nu)\big\vert = \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq (\Phi-\phi)(\Upsilon-\gamma). \end{eqnarray} (3.50)

    From (3.42) and (3.50), we obtain that

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&&\leq\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\Big((\Phi-\phi)(\Upsilon-\gamma)\Big)d\mu d\nu\\&& = \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)(\Phi-\phi)(\Upsilon-\gamma). \end{eqnarray} (3.51)

    This ends the proof.

    Theorem 3.11. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and (1.7) on [0, \infty) and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq L\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\Big)\\ \end{eqnarray} (3.52)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. Taking into consideration the assumption (1.12), we have

    \begin{eqnarray} \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\leq L\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\quad\mu, \nu\in[0, \infty), \end{eqnarray} (3.53)

    which implies that

    \begin{eqnarray} \big\vert\mathfrak{T}(\mu, \nu)\big\vert = \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq L\big( g_{1}(\mu)-g_{1}(\nu)\big)^{2}. \end{eqnarray} (3.54)

    From (3.42) and (3.54), we obtain that

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&&\leq L\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\big( g_{1}(\mu)-g_{1}(\nu)\big)^{2}d\mu d\nu\\&& = L\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1}^{2})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1}^{2})(\varkappa)\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)\Big). \end{eqnarray} (3.55)

    This ends the proof.

    Theorem 3.12. Suppose two integrable functions f_{1} and g_{1} defined on [0, \infty) satisfying the assertions \boldsymbol{I}, \boldsymbol{II} and the lipschitzian condition with the constants \mathcal{M}_{1} and \mathcal{M}_{2} and let two continuous function r and s defined on [0, \infty) . Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \mathcal{M}_{1}\mathcal{M}_{2}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa^{2}s(\varkappa))+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa^{2} r(\varkappa))\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa r(\varkappa))\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa s(\varkappa))\Big)\\ \end{eqnarray} (3.56)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. By the given hypothesis, we have

    \begin{eqnarray} \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\leq \mathcal{M}_{1}\big\vert \mu-\nu \big\vert\quad \Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq \mathcal{M}_{2}\big\vert \mu-\nu \big\vert\quad\mu, \nu\in[0, \infty), \end{eqnarray} (3.57)

    which implies that

    \begin{eqnarray} \big\vert\mathfrak{T}(\mu, \nu)\big\vert = \Big\vert f_{1}(\mu)-f_{1}(\nu) \Big\vert\Big\vert g_{1}(\mu)-g_{1}(\nu) \Big\vert\leq \mathcal{M}_{1}\mathcal{M}_{2}\big( \mu-\nu\big)^{2}. \end{eqnarray} (3.58)

    From (3.42) and (3.58), we obtain that

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}\mathfrak{T}(\mu, \nu)d\mu d\nu\\&&\leq L\frac{\omega^{-2}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)\rho^{\beta}\Gamma(\beta)}\int\limits_{0}^{\varkappa}\int\limits_{0}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]r(\mu)\omega(\mu)\Psi^{\prime}(\mu)}{(\Psi(\varkappa)-\Psi(\mu))^{1-\alpha}}\\&&\quad\times\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\nu))]s(\nu)\omega(\nu)\Psi^{\prime}(\nu)}{(\Psi(\varkappa)-\Psi(\nu))^{1-\beta}}(\mu-\nu)^{2}d\mu d\nu\\&& = \mathcal{M}_{1}\mathcal{M}_{2}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa^{2}s(\varkappa))+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa^{2} r(\varkappa))\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa r(\varkappa))\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa s(\varkappa))\Big). \end{eqnarray} (3.59)

    This ends the proof.

    Corollary 1. Let f_{1} and g_{1} be two differentiable functions on [0, \infty) and let r and s be two non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray} &&\Big\vert\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sf_{1}g_{1})(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\\&&\quad-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(sg_{1})(\varkappa)-\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(sf_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(rg_{1})(\varkappa)\Big\vert\\&&\leq \|f_{1}^{\prime}\|_{\infty}\|g_{1}^{\prime}\|_{\infty}\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa^{2}s(\varkappa))+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa^{2} r(\varkappa))\\&&\quad-2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}(\varkappa r(\varkappa))\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\beta}(\varkappa s(\varkappa))\Big)\\ \end{eqnarray} (3.60)

    holds for all \rho\in(0, 1], \alpha, \beta\in\mathcal{C} with \Re(\alpha), \Re(\beta) > 0.

    Proof. We have f_{1}(\mu)-f_{1}(\nu) = \int\limits_{\nu}^{\mu}f_{1}^{\prime}(\varkappa)d\varkappa and g_{1}(\mu)-g_{1}(\nu) = \int\limits_{\nu}^{\mu}g_{1}^{\prime}(\varkappa)d\varkappa. That is, \big\vert f_{1}(\mu)-f_{1}(\nu)\big\vert\leq\|f_{1}^{\prime}\|_{\infty}\big\vert \mu-\nu \big\vert, \big\vert g_{1}(\mu)-g_{1}(\nu)\big\vert\leq\|g_{1}^{\prime}\|_{\infty}\big\vert \mu-\nu \big\vert, \mu, \nu\in[0, \infty), and the immediate consequence follows from Theorem 3.12. This completes the proof.

    Example 3.13. Let \rho, \, \alpha > 0, \, \, q_{1}, q_{2} > 1 with q_{1}^{-1}+q_{2}^{-1} = 1, and \omega\neq0 be a function on [0, \infty). Let f_{1} be an integrable function defined on [0, \infty) and \, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho; \alpha}f_{1} be the weighted generalized proportional fractional integral operator satisfying assumption \bf{II}. Then we have

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert\leq\Theta\|(f_{1}\circ\omega)(\mu)\|_{L_{1}(a_{1}, \varkappa)}, \end{eqnarray*}

    where

    \begin{eqnarray*} \Theta = \frac{\omega^{-1}(\varkappa)(-1)^{\alpha-1}}{\Gamma(\alpha)}\Big\{\Big(\frac{\rho}{q_{1}(\rho-1)}\Big)^{\alpha-1+{1/q_{1}}}\Big\}^{1/q_{1}}\Phi^{1/q_{1}}\Big(q_{1}(\alpha-1)+1, \frac{q_{1}(\rho-1)}{\rho}\big(\Psi(\varkappa)-\Psi(a_{1})\big)\Big) \end{eqnarray*}

    and

    \Phi(\alpha, \varkappa) = \int\limits_{0}^{\varkappa}e^{-v}v^{\alpha-1}dv

    is the incomplete gamma function [52,53].

    Proof. It follows from Definition 2.2 and the modulus property that

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert\leq\frac{\omega^{-1}(\varkappa)}{\rho^{\alpha}\Gamma(\rho)}\int\limits_{a_{1}}^{\varkappa}\frac{\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]}{\big(\Psi(\varkappa)-\Psi(\mu)\big)^{1-\alpha}}\Psi^{\prime}(\mu)\big\vert f_{1}(\mu)\omega(\mu)\big\vert d\mu \end{eqnarray*}

    for \varkappa > a_{1}.

    Making use of the well-known Hölder inequality, we obtain

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert\leq\frac{\omega^{-1}(\varkappa)}{\rho^{\alpha}\Gamma(\rho)}\Bigg(\int\limits_{a_{1}}^{\varkappa}\frac{q_{1}\exp[\frac{\rho-1}{\rho}(\Psi(\varkappa)-\Psi(\mu))]}{\big(\Psi(\varkappa)-\Psi(\mu)\big)^{q_{1}(1-\alpha)}}\Psi^{\prime}(\mu)d\mu\Bigg)^{1/q_{1}}\| f_{1}\circ\omega(\mu)\|_{L_{1}(a_{1}, \varkappa)}. \end{eqnarray*}

    Let \theta = \Psi(\varkappa)-\Psi(\mu). Then elaborated computations lead to

    \begin{eqnarray*} \Big\vert\Big(\, _{\omega}^{\Psi}\Omega_{a_{1}^{+}}^{\rho;\alpha}f_{1}\Big)(\varkappa)\Big\vert&&\leq\frac{(-1)^{\alpha-1}\omega^{-1}(\varkappa)}{\rho^{\alpha}\Gamma(\alpha)}\Big\{\Big(\frac{\rho}{q_{1}(\rho-1)}\Big)^{\alpha-1+{1/q_{1}}}\Big\}^{1/q_{1}}\nonumber\\&&\quad\times\Phi^{1/q_{1}}\Big(q_{1}(\alpha-1)+1, \frac{q_{1}(\rho-1)}{\rho}\big(\Psi(\varkappa)-\Psi(a_{1})\big)\Big)\| f_{1}\circ\omega(\mu)\|_{L_{1}(a_{1}, \varkappa)}. \end{eqnarray*}

    Here, we aim at present some new generalizations via weighted generalized proportional fractional, weighted generalized Riemann-Liouville and weighted Riemann-Liouville fractional integral operators, which are the new estimates of the main consequences.

    Lemma 4.1. Let f_{1} and g_{1} be two synchronous functions on [0, \infty). Assume that \mathcal{Q} and \mathcal{P} be two non-negative continuous mappings on [0, \infty). Then the inequality

    \begin{eqnarray*} &&\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \Psi(\varkappa) = \varkappa and Lemma 3.1 yields the proof of Lemma 4.1.

    Lemma 4.2. Let f_{1} and g_{1} be two synchronous functions on [0, \infty). Assume that \mathcal{Q} and \mathcal{P} be two non-negative continuous mappings on [0, \infty). Then the inequality

    \begin{eqnarray*} &&\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \Psi(\varkappa) = \varkappa and Lemma 3.1 yields the proof of Lemma 4.2.

    Lemma 4.3. Under the assumption of Lemma 3.1, then the inequality

    \begin{eqnarray*} &&\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1 and Lemma 3.1 yields the proof of Lemma 4.3.

    Lemma 4.4. Under the assumption of Lemma 4.2, then the inequality

    \begin{eqnarray*} &&\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}\big)(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}f_{1}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{Q}\big)(\varkappa)\nonumber\\&&\geq \, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{P}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(\mathcal{P}f_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(\mathcal{Q}g_{1}\big)(\varkappa), \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1, \, \Psi(\varkappa) = \varkappa and Lemma 3.1 yields the proof of Lemma 4.4.

    Theorem 4.5. Let f_{1} and g_{1} be two synchronous functions on [0, \infty). Assume that r, s and t be three non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray*} &&2\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\Big)\nonumber\\&&\quad+2\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}t(\varkappa)\nonumber\\&&\geq\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big) \nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(t g_{1}\big)(\varkappa)\Big)\nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\rho;\alpha}\big(r g_{1}\big)(\varkappa)\Big) \end{eqnarray*}

    holds for all \rho\in(0, 1], \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \Psi(\varkappa) = \varkappa and Theorem 3.2 yields the proof of Theorem 4.5.

    Theorem 4.6. Under the assumption of \boldsymbol{I}, \boldsymbol{II} and let r, s and t be three non-negative continuous functions on [0, \infty). Then the inequality

    \begin{eqnarray*} &&2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa) \, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}t(\varkappa)\Big)\nonumber\\&&\quad+2\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}t(\varkappa)\nonumber\\&&\geq\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big) \nonumber\\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big)\nonumber\\&&\quad+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}^{\Psi}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\Big) \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1 and Theorem 3.2 yields the proof of Theorem 4.6.

    Theorem 4.7. Under the assumption of Theorem 4.5, then the inequality

    \begin{eqnarray*} &&2\, _{\omega}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa) \, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t f_{1}g_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(s f_{1}g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}t(\varkappa)\Big)\nonumber\\&&\quad+2\, _{\omega}\Omega_{0^{+}}^{\alpha}(rf_{1}g_{1})(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}t(\varkappa)\nonumber\\&&\geq\, _{\omega}\Omega_{0^{+}}^{\alpha}r(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big) \nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(rf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(t g_{1}\big)(\varkappa)\Big)\nonumber\\&&\quad+\, _{\omega}\Omega_{0^{+}}^{\alpha}s(\varkappa)\Big(\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(s g_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(r f_{1}\big)(\varkappa)+\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(sf_{1}\big)(\varkappa)\, _{\omega}\Omega_{0^{+}}^{\alpha}\big(r g_{1}\big)(\varkappa)\Big) \end{eqnarray*}

    holds for all \alpha\in\mathcal{C} with \Re(\alpha) > 0.

    Proof. Letting \rho = 1, \, \, \Psi(\varkappa) = \varkappa and Theorem 3.2 yields the proof of Theorem 4.7.

    Remark 5. The computed results lead to the following conclusion:

    (1) Setting \rho = 1, \Psi(\varkappa) = \varkappa and r(\varkappa) = s(\varkappa) = 1, and using the relation (2.7), (2.8) and the assumption \omega(\varkappa) = 1 , then Theorem 3.6 and Theorem 3.9 reduces to the known results due to Dahmani et al. [38].

    (2) Setting \rho = 1, \Psi(\varkappa) = \varkappa and using the relation (2.7), (2.8) and the assumption \omega(\varkappa) = 1 , then Theorem 3.10–3.12, and Corollary 1 reduces to the known results due to Dahmani et al. [38] and Dahmani [40], respectively.

    A new generalized fractional integral operator is proposed in this paper. The novel investigation is used to generate novel weighted fractional operators in the Riemann-Liouville, generalized Riemann-Liouville, Hadamard, Katugampola, Generalized proportional fractional, generalized Hadamard proportional fractional and henceforth, which effectively alleviates the adverse effect of another function \Psi and proportionality index \rho. Utilizing the weighted generalized proportional fractional operator technique, we derived the analogous versions of the extended Chebyshev and Grüss type inequalities that improve the accuracy and efficiency of the proposed technique. Contemplating the Remark 2 and 3, several existing results can be identified in the literature. Some innovative particular cases constructed by this method are tested and analyzed for statistical theory, fractional Schrödinger equation [20,21]. The results show that the method proposed in this paper can stably and efficiently generate integral inequalities for convexity with better operators performance, thus providing a reliable guarantee for its application in control theory [54].

    The authors declare that they have no competing interests.

    The authors would like to express their sincere thanks to referees for improving the article and also thanks to Natural Science Foundation of China (Grant Nos. 61673169) for providing financial assistance to support this research. The authors would like to express their sincere thanks to the support of Taif University Researchers Supporting Project Number (TURSP-2020/217), Taif University, Taif, Saudi Arabia.



    [1] R. Anderson, R. May, Regulation and stability of host-parasite population interactions: I. Regulatory processes, J. Anim. Ecol., 47 (1978), 219–247. doi: 10.2307/3933
    [2] R. Anderson, R. May, Infectious disease of humans: Dynamics and control, Oxford University Press, 1992.
    [3] L. Allen, M. Langlais, C. Phillips, The dynamics of two viral infections in a single host population with applications to hantavirus, Math. Biosci., 186 (2003), 191–217. doi: 10.1016/j.mbs.2003.08.002
    [4] B. Buonomo, D. Lacitignola, C. Leon, Qualitative analysis and optimal control of an epidemic model with vaccination and treatment, Math. Comput. Simulat., 100 (2014), 88–102. doi: 10.1016/j.matcom.2013.11.005
    [5] S. Binder, A. Levitt, J. Sacks, J. Hughes, Emerging infectious diseases: Public health issues for the 21st century, Science, 284 (1999), 1311–1313. doi: 10.1126/science.284.5418.1311
    [6] S. Blower, A. McLean, Mixing ecology and epidemiology, Proc. R. Soc. Lond. B., 245 (1991), 187–192. doi: 10.1098/rspb.1991.0108
    [7] T. Caraballo, M. Fatini, R. Pettersson, R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Cont. Dyn. B, 23 (2018), 2483–3501.
    [8] X. Chen, J. Cao, J. Park, J. Qiu, Stability analysis and estimation of domain of attraction for the endemic equilibrium of an SEIQ epidemic model, Nonlinear Dynam., 87 (2017), 975–985. doi: 10.1007/s11071-016-3092-7
    [9] A. Din, Y. Li, T. Khan, K. Anwar, G. Zaman, Stochastic dynamics of hepatitis B epidemics, Results Phys., 20 (2021), 103730. doi: 10.1016/j.rinp.2020.103730
    [10] A. Din, Y. Li, A. Yusuf, Delayed hepatitis B epidemic model with stochastic analysis, Chaos Soliton. Fract., 146 (2021), 110839. doi: 10.1016/j.chaos.2021.110839
    [11] D. Ebert, C. Zschokke-Rohringer, H. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200–209. doi: 10.1007/PL00008847
    [12] K. Fushimi, Y. Enatsu, E. Ishiwata, Global stability of an SIS epidemic model with delays, Math. Method. Appl. Sci., 41 (2018), 5345–5354. doi: 10.1002/mma.5084
    [13] M. Fatini, M. Khalifi, R. Gerlach, A. Laaribi, R. Taki, Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence, Physica A, 534 (2019), 120696. doi: 10.1016/j.physa.2019.03.061
    [14] X. Feng, L. Liu, S. Tang, X. Huo, Stability and bifurcation analysis of a two-patch SIS model on nosocomial infections, Appl. Math. Lett., 102 (2020), 106097. doi: 10.1016/j.aml.2019.106097
    [15] K. Fan, Y. Zhang, S. Gao, X. Wei, A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity, Physica A, 481 (2017), 198–208. doi: 10.1016/j.physa.2017.04.055
    [16] H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. doi: 10.1137/S0036144500371907
    [17] H. Hethcote, P. Van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271–287. doi: 10.1007/BF00160539
    [18] D. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. doi: 10.1137/S0036144500378302
    [19] M. Jin, Classification of asymptotic behavior in a stochastic SIS epidemic model with vaccination, Physica A, 521 (2019), 661–666. doi: 10.1016/j.physa.2019.01.118
    [20] A. Kumar, M. Kumar, Nilam, A study on the stability behavior of an epidemic model with ratio-dependent incidence and saturated treatment, Theor. Biosci., 139 (2020), 225–234. doi: 10.1007/s12064-020-00314-6
    [21] A. Kumar, Nilam, Stability of a delayed SIR epidemic model by introducing two explicit treatment classes along with nonlinear incidence rate and Holling type treatment, Comput. Appl. Math., 38 (2019), 1–19. doi: 10.1007/s40314-019-0767-y
    [22] W. Kermack, A. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A., 115 (1927), 700–721. doi: 10.1098/rspa.1927.0118
    [23] R. Khasminskii, Stochastic stability of differential equations, Springer Science & Business Media, 2011.
    [24] Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, B. Ahmad, A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Physica A, 551 (2020), 124152. doi: 10.1016/j.physa.2020.124152
    [25] A. Lahrouz, L. Omari, D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model., 16 (2011), 59–76. doi: 10.15388/NA.16.1.14115
    [26] A. Lahrouz, A. Settati, Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation, Appl. Math. Comput., 233 (2014), 10–19.
    [27] H. Li, R. Peng, Dynamics and asymptotic profiles of endemic equilibrium for SIS epidemic patch models, J. Math. Biol., 79 (2019), 1279–1317. doi: 10.1007/s00285-019-01395-8
    [28] J. Li, Y. Yang, Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real, 12 (2011), 2163–2173. doi: 10.1016/j.nonrwa.2010.12.030
    [29] L. Li, Y. Bai, Z. Jin, Periodic solutions of an epidemic model with saturated treatment, Nonlinear Dynam., 76 (2014), 1099–1108. doi: 10.1007/s11071-013-1193-0
    [30] L. Liu, J. Wang, X. Liu, Global stability of an SEIR epidemic model with age-dependent latency and relapse, Nonlinear Anal. Real, 24 (2015), 18–35. doi: 10.1016/j.nonrwa.2015.01.001
    [31] C. Lv, L. Huang, Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response, Commun. Nonlinear Sci., 19 (2014), 121–127. doi: 10.1016/j.cnsns.2013.06.025
    [32] J. Mena-Lorca, H. Hethcote, Dynamic models of infectious disease as regulators of population size, J. Math. Biol., 30 (1992), 693–716.
    [33] H. Moreira, Y. Wang, Global stability in an SIRI model, SIAM Rev., 39 (1997), 496–502. doi: 10.1137/S0036144595295879
    [34] X. Mao, Stochastic differential equations and their applications, Horwood, Chichester, 1997.
    [35] S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135–163. doi: 10.1016/S0022-0396(02)00089-X
    [36] X. Song, A. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281–297. doi: 10.1016/j.jmaa.2006.06.064
    [37] D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev., 32 (1990), 136–139. doi: 10.1137/1032003
    [38] Y. Toshio, On a comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ., 13 (1973), 497–512.
    [39] R. Upadhyay, A. Pal, S. Kumari, P. Roy, Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates, Nonlinear Dynam., 96 (2019), 2351–2368. doi: 10.1007/s11071-019-04926-6
    [40] L. Wang, N. Huang, Ergodic stationary distribution of a stochastic nonlinear epidemic model with relapse and cure, Appl. Anal., 2020 (2020), 1–17.
    [41] World Health Organization, The World Health Report 1996: Fighting disease, Fostering development, World Health Organization, 1996.
    [42] World Health Organization, The world health report 2002: Reducing risks, promoting healthy life, World Health Organization, 2002.
    [43] Q. Yang, X. Mao, Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations, Nonlinear Anal. Real, 14 (2013), 1434–1456. doi: 10.1016/j.nonrwa.2012.10.007
    [44] Y. Yang, J. Zhou, C. H. Hsu, Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate, J. Math. Anal. Appl., 478 (2019), 874–896. doi: 10.1016/j.jmaa.2019.05.059
    [45] Z. Zhang, Y. Suo, Qualitative analysis of a SIR epidemic model with saturated treatment rate, J. Appl. Math. Comput., 34 (2010), 177–194. doi: 10.1007/s12190-009-0315-9
    [46] Y. Zhao, D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718–727.
  • This article has been cited by:

    1. Mohammed Shehu Shagari, Qiu-Hong Shi, Saima Rashid, Usamot Idayat Foluke, Khadijah M. Abualnaja, Fixed points of nonlinear contractions with applications, 2021, 6, 2473-6988, 9378, 10.3934/math.2021545
    2. Farhat Safdar, Muhammad Attique, Some new generalizations for exponentially (s, m)-preinvex functions considering generalized fractional integral operators, 2021, 1016-2526, 861, 10.52280/pujm.2021.531203
    3. Shuang-Shuang Zhou, Saima Rashid, Erhan Set, Abdulaziz Garba Ahmad, Y. S. Hamed, On more general inequalities for weighted generalized proportional Hadamard fractional integral operator with applications, 2021, 6, 2473-6988, 9154, 10.3934/math.2021532
    4. Saima Rashid, Aasma Khalid, Omar Bazighifan, Georgia Irina Oros, New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications, 2021, 9, 2227-7390, 1753, 10.3390/math9151753
    5. Saima Rashid, Fahd Jarad, Khadijah M. Abualnaja, On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfer-generalized proportional fractional derivative, 2021, 6, 2473-6988, 10920, 10.3934/math.2021635
    6. SAIMA RASHID, ELBAZ I. ABOUELMAGD, AASMA KHALID, FOZIA BASHIR FAROOQ, YU-MING CHU, SOME RECENT DEVELOPMENTS ON DYNAMICAL ℏ-DISCRETE FRACTIONAL TYPE INEQUALITIES IN THE FRAME OF NONSINGULAR AND NONLOCAL KERNELS, 2022, 30, 0218-348X, 10.1142/S0218348X22401107
    7. Wengui Yang, Certain New Chebyshev and Grüss-Type Inequalities for Unified Fractional Integral Operators via an Extended Generalized Mittag-Leffler Function, 2022, 6, 2504-3110, 182, 10.3390/fractalfract6040182
    8. Shuang-Shuang Zhou, Saima Rashid, Asia Rauf, Khadija Tul Kubra, Abdullah M. Alsharif, Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time, 2021, 6, 2473-6988, 12114, 10.3934/math.2021703
    9. Songkran Pleumpreedaporn, Chanidaporn Pleumpreedaporn, Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Jehad Alzabut, On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function, 2022, 7, 2473-6988, 7817, 10.3934/math.2022438
    10. SAIMA RASHID, AASMA KHALID, YELIZ KARACA, YU-MING CHU, REVISITING FEJÉR–HERMITE–HADAMARD TYPE INEQUALITIES IN FRACTAL DOMAIN AND APPLICATIONS, 2022, 30, 0218-348X, 10.1142/S0218348X22401338
    11. Bounmy Khaminsou, Weerawat Sudsutad, Jutarat Kongson, Somsiri Nontasawatsri, Adirek Vajrapatkul, Chatthai Thaiprayoon, Investigation of Caputo proportional fractional integro-differential equation with mixed nonlocal conditions with respect to another function, 2022, 7, 2473-6988, 9549, 10.3934/math.2022531
    12. Fuxiang Liu, Jielan Li, Analytical Properties and Hermite–Hadamard Type Inequalities Derived from Multiplicative Generalized Proportional σ-Riemann–Liouville Fractional Integrals, 2025, 17, 2073-8994, 702, 10.3390/sym17050702
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3254) PDF downloads(172) Cited by(15)

Figures and Tables

Figures(3)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog